Lecture notes: Reversible computing with probability amplitudes

Transcription

1 Phys 711 Topics in Particles & Fields Spring 013 Lecture 4 v Lecture notes: Reversible computing with probability amplitudes Jeffrey Yepez Department of Physics and Astronomy University of Hawai i at Manoa Watanabe Hall, 505 Correa Road Honolulu, Hawai i yepez@hawaii.edu yepez Dated: January 30, 013 Contents I. Introduction 1 II. Classical gas in a confining lattice III. Unitary algorithmic representation of the Schroedinger equation 3 IV. Final Remarks 6 References 7 I. INTRODUCTION Perhaps the earliest unitary algorithm for approximating a nonrelativistic scalar quantum particle on a computer is the Cayley form of the evolution operator for the Schroedinger wave equation Goldberg et al., In 1994 Bialynicki-Birula presented a general quantum algorithmic approach of this kind for modeling the Weyl, Dirac, and Maxwell equations on a body-centered cubic lattice in three-dimensions Bialynicki-Birula, This model is 1st-order convergent doubling the grid resolution merely doubles the numerical accuracy, but convergence is problematic when modeling particle dynamics in an external potential. Although the model is unitary, it is specified using non-unitary matrices and requires ad hoc lattice partitioning if implemented in parallel. Furthermore, Bialynicki-Birula addresses only the one-body problem. In a series of papers on simulating the one-dimensional Dirac equation Meyer, 1996, 1997a,b, Meyer presented a quantum algorithm similar to that of Bialynicki-Birula with a variety of numerical simulations including the effects of boundary conditions, inhomogeneities, and an external scalar potential. Meyer set the quantum algorithm for the discretized path integral in the context of the quantum lattice gas method, and his algorithm is equivalent to the one-dimensional version of the Bialynicki- Birula quantum algorithm for the Dirac equation. Meyer contributed 1D one-body numerical simulations and addressed the non-interacting lattice or checkerboard problem using an additional rest particle. Yet he too did not address the many-body case nor the low-order numerical convergence issue. The approach taken by Succi and Benzi is a classicallyoriented computational approach that begins with a kinetic lattice Boltzmann equation of motion 1 effectively the one-dimensional Dirac equation in the Majorana representation and shows that the Schroedinger wave equation emerges as the governing equation of motion for the slow mode in the long wavelength hydrodynamic limit. That is, Succi and Benzi observed that the Schroedinger wave equation arises from the Dirac equation in a manner quite analogous to how the macroscopic Navier-Stokes hydrodynamic fluid equation arises from the mesoscopic kinetic Boltzmann equation through the Chapman-Enskog expansion. Succi published a series of papers on this subject emphasizing the analogy between quantum mechanics and fluid mechanics: the connection between the Dirac equation and the Schroedinger equation to that between the kinetic Boltzmann equation and the Navier-Stokes equation of hydrodynamics Succi, 1996, 1998; Succi and Benzi, Succi s quantum lattice gas model on a cubic lattice for the 3D Dirac equation has, at the kinetic level, the particles undergoing mixing during free propagation and is again similar to Bialynicki-Birula s model. Succi discusses the many-body case, but his algorithm runs into an exponential complexity wall in this case Succi, 00 because it is still implemented as a set of one-body systems at the mesoscopic scale. Boghosian and Taylor followed an approach along the lines of Meyer s approach in that their model is developed as a generalization of the classical lattice gas method. A quantum spin system, now formulated directly at a small scale, again leads to the Schroedinger wave equation in the continuum limit. The Boghosian and Taylor quantum lattice gas model focuses on solving the many-body Schroedinger wave equation with an arbitrary scalar potential in an arbitrary number of spatial dimensions. They argue that an exponential numerical speedup arises from simulation in the many-body sector of the full Hilbert space carried out simultaneously us- 1 The classical lattice Boltzmann equation became well known for its application to computational fluid dynamics Chen et al., 199; Qian et al., 199; Succi et al., 1991.

2 II CLASSICAL GAS IN A CONFINING LATTICE ing quantum superposition of states. The Boghosian and Taylor version of the quantum algorithm is cast explicitly for direct implementation using an array of quantum bits Boghosian and Taylor, Polley has presented an argument for inserting both an external scalar and vector potential into a quantum lattice-gas model by analytically demonstrating the discrete model s invariance with respect to a general local gauge transformation Polley, 000. Zalka proposed a quantum algorithm to model the Schroedinger wave equation with an external scalar potential Zalka, Aug A periodic wave function is mapped onto the numbered basis states of the full Hilbert space of the quantum computer. The numerous Hilbert space dimensions are mapped onto the many coordinates of a quantum system of one spatial dimension. Zalka s approach does not offer a way to measure the time-dependent behavior of the modeled quantum system, because upon observation of the l-bit quantum register the l number of amplitudes associated with the modeled wave function s values at each spatial coordinate collapse into only l classical bits. Here we present a unitary algorithm for representing the behavior of a quantum particle governed by the Schroedinger wave equation, where the form of the unitary algorithm was found by generalizing the reversible algorithm for a classical gas confined to a lattice Yepez, 1996c. The first application of this particular unitary algorithm was to model Laudau s two fluid equations for a superfluid Yepez, 1996b. The second application was to model normal fluid dynamics Yepez, 1999b. In Sec. II we present a summary of the derivation of the kinetic transport equation for a classical gas confined to a lattice. In Sec. III we review an original formulation that has the merit of being quite simple albeit only first-order accurate in its numerical convergence. A revised treatment of this quantum algorithm that increases the numerical accuracy to second-order is considered in Problem 4 the qubit representation useful for manybody simulations appears in Refs. Yepez, 007; Yepez and Boghosian, 00. II. CLASSICAL GAS IN A CONFINING LATTICE In the previous lecture on reversible computing, we represented the reversible particle dynamics of a classical gas in a lattice by a discrete Boltzmann equation at the mesoscopic scale, which for our purposes here we write as fx + le, t + τ = fx, t + Ω meso fx, t, 1a where the components of the kinetic-space ket fx, t are the occupancy probabilities associated with the single-particle distribution function along the lattice directions f 1 x, t f x, t fx, t., f B x, t 1b where the displacement le technically a 3-vector of matrices of size B B is implicitly defined as f 1 x + le 1, t fx + le, t = e le f x + le, t fx, t., f B x + le B, t 1c and where the lattice vectors are e a for a [1, B] for lattice coordination number B. If we expand the kineticspace ket in the order of its fluctuations fx, t = f 0 x, t + f 1 x, t +, then the collision function is Ω meso f = J f 1 +, where J is a circulant matrix. J is the Jacobian of the collision function discussed in the last lecture. Taylor expanding 1a up to second order fluctuations and dividing by the time step τ gives the kinetic transport equation t f + l l e f + τ τ e f + = 1 τ J f Inserting into 3, and counting a spatial derivative as a first order fluctuation and a time derivative as second order, gives the following first and second order kinetic transport equations: l τ e f 0 = 1 τ J f 1 4a t f 0 + l τ e f 1 + l τ e f 0 = b Solving 4a for f 1 and inserting this into 4b gives an effective low-frequency and small-wave number field equation of motion for the dynamical local equilibrium at the kinetic level t f 0 + l τ e J 1 e + 1 e f 0 = As an example of the displacements e, consider a classical gas confined to a two-dimensional lattice with the following B = 4 lattice vectors: e 1 = ŷ + ẑ 6a e = ŷ ẑ 6b e 3 = ŷ ẑ 6c e 4 = ŷ + ẑ, 6d

3 III UNITARY ALGORITHMIC REPRESENTATION OF THE SCHROEDINGER EQUATION one per quadrant. The length of the lattice vectors is, so the hopping speed in the model is c = l/τ. In turn, the acoustic sound speed for this classical gas, c s = c/ D, is unity in lattice units for D =. Thus, we have f 1 x + lŷ + lẑ fx + le, t = e le f fx, t = x + lŷ lẑ f 3 x lŷ lẑ, f 4 x lŷ + lẑ 7a or equivalently e = σ z 1 ŷ + σ z σ z ẑ. 7b Now we seek a similar but more general representation of particle dynamics we consider a quantum gas in a confining lattice Yepez, 1996b,c, 1999b. To construct the quantum model, we use the mapping probabilities probability amplitudes 8 real-valued e complex-valued e 9 bits qubits. 10 So a quantum generalization of 1a is a discrete quantum Boltzmann equation ψx + le, t + τ = ψx, t + Ω qu ψx, t, where the local quantum state ψ 1 x, t ψ x, t ψx, t. ψ B x, t 11a 11b has c-number components chosen such that its values encode the occupancy probabilities according to the prescription f a x, t ψ ax, tψ a x, t, for a [1, B]. To represent particle dynamics in three spatial dimensions one may choose the displacements to be e = σ z σ x ˆx + σ z σ y ŷ + σ z σ z ẑ. 11c The effective field theory that describes the quantum lattice gas field 11 is derived in the next section. The quantum state 11b resides in the single-particle subspace of the Hilbert space. The single-particle subspace is spanned by the numbered states 1000, 0100, 0010, and 0001 at point x. That is, with B = 4, there are four qubits per point encoding the probability amplitudes according to the prescription q a x, t = ψ a x, t ψ a x, t 0. 1 Employing the number operators n a = 1 a 1 n, we may write the single-particle occupancy probabilities as f a = ψx, t n a ψx, t, 13 which is real-valued because the number operators are diagonal. We can interpret ψx, t as a spinor-valued quantum state. In this lecture we will consider the simplest model that represents the dynamics of a free single quantum particle and derive the required quantum unitary collision operator needed for this purpose. Remarkably, the same unitary operator also represents the dynamics of a many-body system of free quantum particles, which is the source of the efficiency of the quantum algorithm. Exercise: If it is not already obvious, for the B = 4 example verify that the occupancy probability prescription f a x, t = ψ ax, tψ a x, t comes from inserting the number operators n 1 = n n = 1 n 1 1 n 3 = 1 1 n 1 n 4 = n into f a x, t = ψx, t n a ψx, t when the ket ψx, t resides in the single-particle subspace of the Hilbert space of the quantum node at x. III. UNITARY ALGORITHMIC REPRESENTATION OF THE SCHROEDINGER EQUATION Complex amplitudes denoted by ψ s ς, for s = L, R and ς =,, reside at each site of the lattice, where the position of the lattice site is specified by the vector x i, for i = 1,, 3. At each spacetime point x = t, x, a 4-spinor denoted ψ has these amplitudes as its components ψx = ψ L x ψ L x ψ R x ψ R x. 14 To describe the dynamics, let us begin with the streaming operators, which are related to the Lorentz group boosts, and these directly act on the chirality-spin degrees of freedom in 14. The generators for boosts are κ i = σ 3 σ i, 15 Here we do not choose to use the notation ψx x ψ, conventionally used in quantum mechanics for scalar wave functions cast as the projection of a quantum ket onto a position-basis bra. 3

4 III UNITARY ALGORITHMIC REPRESENTATION OF THE SCHROEDINGER EQUATION where σ i are the Pauli matrices, again for i = 1,, 3. The stream operators expressed in unitary exponential form are Υ S i ζ = e iζ κi = 1 + i sinζκ i + cosζ 1κ i. 16 These stream operators preserve chirality. Note that the third generator of 15 is diagonal κ 3 = σ 3 σ 3 = , 17 so as a matter of algorithmic practicality, we implement all the stream operators strictly using 17. That is, the quantum algorithm s stream operators along the x and y directions may be diagonalized and so they can be specified by Υ S 1ζ = e i π 4 1 σy Υ S 3ζ e i π 4 1 σy Υ S ζ = e i π 4 1 σx Υ S 3ζ e i π 4 1 σx. 18a 18b With p = i, the basic quantum lattice gas stream operator acting on the 4-spinor gives Υ S 3δx p/ ψx = e +δx ψ L x e δx ψ L x e δx ψ R x e +δx ψ R x, 19 amounting to a shift of the amplitudes for the 4-spinor components. This is equivalent to a simple permutation between the points of the space a unitary operation of the amplitudes Eq. 19 S δx ψx = ψ L x + δx, t ψ L x δx, t ψ R x δx, t ψ R x + δx, t. 0 Furthermore, in any quantum computational implementation, the shift permutations 0 are represented in terms of an ordered sequence of qubit-qubit interchange gates Yepez, 001. Streaming of all the 4-spinor amplitudes along the x, y, and z directions on a cubical lattice is represented by the composite operator S = x = x Υ S 1δx p x / Υ S δy p y / Υ S 3δz p z / 1a [ ] e i π 4 1 σy Υ S 3δx p x / e i π 4 1 σy+σx Υ S 3δy p y / e i π 4 1 σx Υ S 3δz p z /, 1b where the grid sizes are a fixed length, δx = δy = δz = εl. Streaming represents the high-energy chiral motion Yepez, 003 by the following map: ψ = S ψ = e εl σ3 σ ψ ψ. Alternatively, the stream equation can be written exactly as a grid-level difference equation ψx + εl σ 3 σ, t + ετ = ψx, t. 3 In the low-energy scaling limit, obtained by Taylor expanding the spinor field in space and time under advective ordering i.e. δt δx ε, this in turn leads to the effective dynamical equation of motion t ψx + = l τ σ 3 σ ψx +, 4 which is just the relativistically covariant Weyl equation for a 4-spinor iγ µ µ ψ = 0, 5 written in the chiral representation where the Dirac matrices are γ 0 = σ 1 1 and γ i = iσ σ i, and where the hopping speed is c l/τ = 1 in natural units. Next, let us consider the unitary operator U = e ihgasτ/, 6 where h gas = ωn, and where N is an involution operator N = 1 that is dimensionless and represented by a hermitian matrix e.g. N = σ x. Thus, we can write U = 1 cos ωτ i N sin ωτ = 1 + J, 7 where J cos ωτ 1 1 i N sin ωτ and where 1 is the identity matrix. In a quantum gate representation of 7, the gate angle is ωτ. We employ a unitary collision operator C U 1 to locally scatter the incoming 4-spinor ψx to an outgoing 4-spinor ψ x : ψ x = C ψx. 8 4

5 III UNITARY ALGORITHMIC REPRESENTATION OF THE SCHROEDINGER EQUATION C breaks chirality and imparts mass to the Weyl particle represented by the stream operator. The quantum algorithm we consider is the combination of and 8 ψ = S C ψ ψ, 9 which can be written exactly as the following grid-level transport equation 3 ψx + εlσ z σ, t + ε τ 7 = ψx, t + J 1 ψx, t, 30 3 We are using diffusive ordering where δt δx ε. 4 Equation 30 was written in a form akin to the lattice Boltzmann equation by using a complex-valued distribution encoded and this is an exact expression of the quantum algorithm. 4 Equation 30 is referred to as a quantum lattice gas equation of motion. The collide and stream based evolution 9 constitutes the simplest quantum lattice gas algorithm to model a free massive quantum particle in position-space in the long wavelength limit. Taylor expanding the left-hand side of 30 in space gives the associated local difference equation in time, to second order in ε, as the components of ψ in place of a real-valued probability distribution encoded in the components of f Yepez, 1996c. J 1 ψx, t = ψx, t + ε τ ψx, t + εlσ z σ + 1 εlσ z σ ψx, t + ε τ Let us perform a perturbative expansion of the 4-spinor in the small dimensionless parameter ε as follows: ψ = ψ 0 + ψ 1 + ψ +, 3 where ψ 1 ε, ψ ε, and so forth. Inserting this into 31 and equating terms of similar order in ε gives the following zeroth, first, and second order fluctuations 5 J 1 ψ 0 t = 0 33a 5 At zeroth order the operator J does not affect ψ 0 since this is the equilibrium state i.e. J ψ 0 = 0. J 1 ψ 1 t = εlσ z σ ψ 0 t + ε τ 33b ψ 0 t + ε τ ψ 0 t = εlσ z σ ψ 1 t + ε τ ε lσ z σ ψ 0 t + ε τ. 33c Equation 33b can be inverted to solve for the first order correction to the 4-spinor field, ψ 1. Substituting this into 33c, a difference equation for ψ 0 emerges which reduces to ψ 0 t + ε τ ψ 0 t = ε l σ z σ J 1 1 σ z σ : ψ 0 t + ε τ ε l σ z σ ψ 0 t + ε τ, τ t ψ 0 = l σ z J 1 σ z σ ψ 0 l σ z σ ψ 0 +, 34a 34b where we have expanded both the left-hand and righthand sides and retained only the relevant order-ε terms. This further simplifies since σ =, an identity that follows from the anticommutivity of the Pauli matrices. So, we arrive at the effective 4-spinor equation of motion t ψ 0 = l τ σ z J 1 σ z ψ

6 IV FINAL REMARKS This then separates into a pair of -spinor equations ψ Lς 0 t = l σ z J 1 σ z + 1 ψ Lς 0, 36 τ ψ Rς 0 ψ 0 Rς one equation for each of the spin-up and spin-down components, ς =,. Now, we consider the equation of motion for the - spinor field ψ 0 = ψ 0 L ψ 0 R 37 where here, for convenience, we suppress the spin index. The application of the unitary collision operator U = 1+ J may cause large changes in the -spinor at every time step iteration. To avoid this, it is possible to transform to a reference frame where the spinor is always near local equilibrium. Denote the spinor field in this frame by η and the transformation matrix by R, so that η R ψ. The requirement for being in the local equilibrium frame is that the unitary collision operator in that frame, C = 1 + J, be diagonal. The grid-level quantum lattice gas equation in the local equilibrium frame becomes ηx + εσ z, t + ε τ = C ηx, t, 38 where C is diagonalized by the similarity transformation C = RCR 1. Therefore, in the local equilibrium frame, 35 is diagonalized, becoming a parabolic partial differential equation in space and time for each of its -spinor components, independently, t η 0 = l τ R σ z J σ z R 1 η 0 t. 39 Consider the following choice for the similarity transformation 6 R = The matrix on the right hand side of 39 must be diagonal, so σ z J σz must be off-diagonal. There are different choices for the form of J 1 available to use. 7 The original choice Yepez, 1996c was the following: J 1 = 1 1 i mν i, 41 mν 1 which when inverted gives J = 1 + m ν 1 i mν i mν 1, 4 and this in turn gives the unitary matrix 1 U = 1 + J = m ν 1 i mν 1 + i. 43 m ν mν m ν 1 Note that /mν is a dimensionless real-valued number. That is, with these choices 40 and 43 for the similarity transformation matrix, R, and the unitary matrix, U, 35 reduces to i t ψ 0 = m τν σ x ψ In turn, in the local equilibrium frame, 39 becomes l l i t η 0 = m τν σ z η 0 +, 45 and letting η 0 u v T, we have i t u = m u + i t v = m v +, 46a 46b where we have taken ν = l /τ. Since η = R ψ, we ψ have η = 1 L + ψ R. In 46, the bosonic amplitude ψ L ψ R field u = ψ L + ψ R / obeys the Schroedinger equation while v = ψ L ψ R / obeys the time-reversed Schroedinger equation. Hence, it is possible to recover the quantum wave equations from a quantum lattice gas in analogy to the recovery of the Navier-Stokes equation from a classical lattice gas Yepez, 1996a, 1999a, which was the original motivation that led to the discovery of this algorithm. 6 Note R = R 1 and it has the property that it transforms an off-diagonal matrix into a diagonal matrix!! R 0 1 R = A typical choice is U = e iθσx, so that J = This has as its inverse J 1 = 1 cos θ 1 1 i cot θ i cot θ i sin θ cos θ 1 1 «i sin θ. «. Hence, the determinative term on the right-hand side of 36 is σ 3 `J σ3 = i σ 1 cot θ. So, the gate angle θ sets the particle mass, m = tan θ. IV. FINAL REMARKS In this lecture we reviewed a quantum lattice gas algorithm for the free Schroedinger equation in 3+1 dimensions. One aim was to explain the framework for deriving an effective equation of motion starting from a given unitary evolution that is separated into a product of two fundamental operations: 1 streaming that represents the kinetic motion of the quantum particle and collisions that represent local scattering events or reversal of motion that impart mass to the quantum particle. 6

7 IV FINAL REMARKS We began with a discrete kinetic-level treatment of a classical gas on a lattice to provide the motivation for a spinor representation in a discrete spacetime. We postpone the discussion of the qubit representation of a quantum field to focus on the basic notion of unitary evolution a technique very important to state-of-the-art computational physics applied to the numerical simulation of quantum systems. The quantum algorithm we presented has error terms in the effective equation of motion that occur at first-order. This is not too practical for quantum simulation purposes because to reduce the numerical error by a factor of two one must likewise decrease the grid size by a factor of two; on a cubical grid this increases the number of points by 3, which is a large nearly an order of magnitude increase in computational burden. Yet, our aim was to derive the effective equation of motion in 3+1 dimensions in the simplest way possible with a minimal construction based on a reversible classical gas confined to a lattice. A version of this quantum algorithm that has error terms in its effective equation of motion at fourth-order is presented in Problem 4, and this quantum algorithm is practical for quantum simulation purposes. The material in this lecture originally appeared in Ref. Yepez, 1996b,c Ref. Yepez, 1996b is reprinted as an Appendix to Lecture 4 since it is unpublished. The streaming operator used for the 3+1 dimensional model originally appeared in Ref. Yepez, 003. Yepez, J., 1996c, USAF Technical Report DTIC ADA4171, URL ADA4171. Yepez, J., 1999a, The classical lattice-gas method, Technical Report, Air Force Research Laboratory, AFRL/RV Hanscom AFB, MA 01731, DTIC ADA Yepez, J., 1999b, in Quantum Computing and Quantum Communications, edited by C. P. Williams, Lecture Notes in Computer Science Springer-Verlag, p. 480pp, First NASA International Conference, QCQC 98, Palm Springs, California, USA, February 17-0, 1998, Selected Papers. Yepez, J., 001, Physical Review E 634, Yepez, J., 003, arxiv.org e-print archive Quant-ph/ Yepez, J., 007, Lattice Gas Dynamics: Volume III Quantum algorithms for computational physics, Technical Report AFRL-VS-HA-TR , Air Force Research Laboratory, AFRL/RV Hanscom AFB, MA 01731, DTIC ADA Yepez, J., and B. Boghosian, 00, Computer Physics Communications 1463, 80. Zalka, C., Aug 1996, Los Alamos National Laboratory Archive quantum-ph960306v, 8 pages. References Bialynicki-Birula, I., 1994, Physical Review D 491, 690. Boghosian, B. M., and W. Taylor, 1998, Physica D 10, 30. Chen, H., S. Chen, and W. H. Mattaeus, 199, Physical Review A 458, R5339. Goldberg, A., H. M. Schey, and J. L. Schwartz, 1967, American Journal of Physics 353, 177, URL org/link/?ajp/35/177/1. Meyer, D. A., 1996, Journal of Statistical Physics 855,6, 551. Meyer, D. A., 1997a, Quant-Ph 97105, 1. Meyer, D. A., 1997b, Physical Review E 555, 561. Polley, L., 000, Los Alamos National Laboratory Electronic Archive quant-ph/ , 1. Qian, Y., D. d Humiéres, and P. Lallemand, 199, Europhysics Letters 176BIS, 479. Succi, S., 1996, Physical Review E 53, Succi, S., 1998, International Journal of Modern Physics C 98, Succi, S., 00, Computer Physics Communications 1463, 317. Succi, S., and R. Benzi, 1993, Physica D 69, 37. Succi, S., R. Benzi, and F. Higuera, 1991, Physica D 47, 19. Yepez, J., 1996a, Lattice Gas Dynamics: Volume 1 Viscous Fluids, Technical Report PL-TR-96-1I, Air Force Research Laboratory, AFRL/VSBE Hanscom AFB, MA 01731, DTIC ADA Yepez, J., 1996b, Unpublished Cashed on CiteSeer doi: Presented at the Sixth International Conference on Discrete Models for Fluid Mechanics, Boston University Center for Computational Science, August